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On a McCoy-like condition for rings

Sharang Thimmaiah, Raisa DSouza

Abstract

We study rings $R$ for which whenever non-zero polynomials $f(x)$ and $g(x)$ satisfy $f(x)g(x)f(x)=0$, it implies that there is a non-zero element $r\in R$ such that $f(x)rf(x)=0$. We call such rings inner McCoy rings. We explore some examples of rings that are inner McCoy, determine relationships between the class of inner McCoy rings and some known classes of rings. Furthermore, we show that the maximal inner McCoy subring of a matrix ring is the triangular matrix ring. Finally, we construct new inner McCoy rings from known ones.

On a McCoy-like condition for rings

Abstract

We study rings for which whenever non-zero polynomials and satisfy , it implies that there is a non-zero element such that . We call such rings inner McCoy rings. We explore some examples of rings that are inner McCoy, determine relationships between the class of inner McCoy rings and some known classes of rings. Furthermore, we show that the maximal inner McCoy subring of a matrix ring is the triangular matrix ring. Finally, we construct new inner McCoy rings from known ones.
Paper Structure (5 sections, 14 theorems, 22 equations)

This paper contains 5 sections, 14 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Inner McCoy Rings
  3. Semi-commutative, i-reversible and Inner McCoy Rings
  4. Maximal inner McCoy subring of the full matrix ring
  5. Some Extensions of Inner McCoy Rings

Key Result

Lemma 1.1

mccoy1942 Suppose $R$ is a commutative ring with unity. Let $f(x)$ and $g(x)$ be two nonzero elements of the ring $R[x]$ such that $f(x)g(x)=0$. Then, we can find nonzero elements $s, t \in R$ such that $sg(x) = 0$ and $f(x)t = 0$.

Theorems & Definitions (37)

  • Lemma 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • ...and 27 more